Exchange Enhancement of the Electron-Phonon Interaction: the Case of Weakly Doped Two-Dimensional Multivalley Semiconductors

The effect of the exchange interaction on the vibrational properties and on the electron-phonon coupling were investigated in several recent works. In most of the case, exchange tends to enhance the electron-phonon interaction, although the motivations for such behaviour are not completely understood. Here we consider the class of weakly doped two-dimensional multivalley semiconductors and we demonstrate that a more global picture emerges. In particular we show that in these systems, at low enough doping, even a moderate electron-electron interaction enhances the response to any perturbation inducing a valley polarization. If the valley polarization is due to the electron-phonon coupling, the electron-electron interaction results in an enhancement of the superconducting critical temperature. We demonstrate the applicability of the theory by performing random phase approximation and first principles calculations in transition metal chloronitrides. We find that exchange is responsible for the enhancement of the superconducting critical temperature in Li$_x$ZrNCl and that much larger T$_c$s could be obtained in intercalated HfNCl if the synthesis of cleaner samples could remove the Anderson insulating state competing with superconductivity.